This discussion is confined to one dimensional (1D) filters because the N dimensional case is readily generalized from the 1D case.
A 1D filter is characterized by a vector of possibly complex valued weights or parameters. These weights are the tap values in a transversal filter arrangement. These weights are derived from the discrete approximation to the continuous convolution of an input signal and the impulse response of the system to filter. The convolution characterizes the filter's impact on the input signal, as follows:
      y    ⁡          (      t      )        =            ∫              -        ∞                    +        ∞              ⁢                  h        ⁡                  (                      t            -            τ                    )                    ⁢              u        ⁡                  (          τ          )                    ⁢                          ⁢              ⅆ        τ            where u(τ) is the input signal and h(t−τ) is the (non-causal in this case) impulse response of the system shown in FIG. 1.
The impulse response of a system is the result of applying a sharp spiked signal at the input. The spiked signal is a discrete approximation to the unrealizable impulse of zero duration and infinite amplitude (defined using the Dirac delta functional). The idealized spectrum of impulse is flat with unit amplitude and constant phase.
The discrete approximation of the impulse response is:
      y    ⁡          (      t      )        =                    ∫                  -          ∞                          +          ∞                    ⁢                        h          ⁡                      (                          t              -              τ                        )                          ⁢                  u          ⁡                      (            τ            )                          ⁢                                  ⁢                  ⅆ          τ                      ≈                  1        /        N            ⁢                        ∑                      k            =            0                                k            =                          N              -              1                                      ⁢                                  ⁢                              h            ⁡                          (                              n                -                k                            )                                ⁢                      u            ⁡                          (              k              )                                          where 1/N is the uniform, Nyquist, sampling rate used to digitize the continuous signal.
Digital filters for linear systems are categorized as Finite-Impulse Response (FIR) or Infinite Impulse Response (IIR) in architecture. The IIR occurs when the output at one sample time is fed back as input to the next sample event. FIGS. 2 and 3 depict a general purpose FIR or IIR digital filter, respectively, with both architected as a transverse filter.
There are many varied techniques for designing digital filters. Static filter design entails the determination of static tap weights for specific problem domains, bounded by requirements and resource constraints. Adaptive digital filters result when the tap weights are allowed to vary, typically depending on a comparison of the actual output to a reference output. Typically, the Weiner-Hopf equation provides the analytical basis for calculating an adaptive filter's optimal tap weights.